Ambiguous Grammars - ByteLABS

Ambiguous Grammars
Definitions
• If a grammar has more than one leftmost derivation for a
single sentential form, the grammar is ambiguous
• If a grammar has more than one rightmost derivation for a
single sentential form, the grammar is ambiguous
• The leftmost and rightmost derivations for a sentential
form may differ, even in an unambiguous grammar
— However, they must have the same parse tree!
Classic example — the if-then-else problem
Stmt → if Expr then Stmt
| if Expr then Stmt else Stmt
| … other stmts …
This ambiguity is entirely grammatical in nature
COMP 412, Fall 2002 16
Comp 412 Fall 2005

Ambiguity
This sentential form has two derivations
if Expr 1
then if Expr 2
then Stmt 1
else Stmt 2
if
if
E 1
then
else
E 1
then
if
S 2
E 2
then
if
E 2
then else
S 1
S 1
S 2
production 2,
then production 1
production 1,
then production 2
COMP 412, Fall 2002 17
Comp 412 Fall 2005

Ambiguity
Removing the ambiguity
• Must rewrite the grammar to avoid generating the problem
• Match each else to innermost unmatched if (common sense rule)
1
Stmt
→
if Expr then Stmt
2
⏐
if Expr then WithElse else Stmt
3
⏐
Other Statements
4
WithElse
→
if Expr then WithElse else WithElse
5
⏐
Other Statements
With this grammar, example has only one rightmost derivation
Intuition: once into WithElse, we cannot generate an unmatched else
… a final if without an else can only come through rule 2 …
COMP 412, Fall 2002 18
Comp 412 Fall 2005

Ambiguity
if Expr 1 then if Expr 2 then Stmt 1 else Stmt 2
Rule Sentential Form
— Stmt
1 if Expr then Stmt
2 if Expr then if Expr then WithElse else Stmt
3 if Expr then if Expr then WithElse else S 2
5 if Expr then if Expr then S 1 else S 2
? if Expr then if E 2 then S 1 else S 2
? if E 1 then if E 2 then S 1 else S 2
some other production
This grammar has only one rightmost derivation for the example
COMP 412, Fall 2002 19
Comp 412 Fall 2005